Theorem natoppfb 49193

Description: A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
natoppf.o	⊢ 𝑂 = (oppCat‘𝐶)
natoppf.p	⊢ 𝑃 = (oppCat‘𝐷)
natoppf.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
natoppf.m	⊢ 𝑀 = (𝑂 Nat 𝑃)
natoppfb.k	⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
natoppfb.l	⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
natoppfb.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
natoppfb.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
natoppfb	⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Proof of Theorem natoppfb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		natoppf.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
2		natoppf.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐷)
3		natoppf.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		natoppf.m	. . . 4 ⊢ 𝑀 = (𝑂 Nat 𝑃)
5		natoppfb.k	. . . . 5 ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐾 = ( oppFunc ‘𝐹))
7		natoppfb.l	. . . . 5 ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐿 = ( oppFunc ‘𝐺))
9		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐹𝑁𝐺))
10	1, 2, 3, 4, 6, 8, 9	natoppf2 49192	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐿𝑀𝐾))
11		eqid 2729	. . . . 5 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
12		eqid 2729	. . . . 5 ⊢ (oppCat‘𝑃) = (oppCat‘𝑃)
13		eqid 2729	. . . . 5 ⊢ ((oppCat‘𝑂) Nat (oppCat‘𝑃)) = ((oppCat‘𝑂) Nat (oppCat‘𝑃))
14	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 = ( oppFunc ‘𝐺))
15	14	fveq2d 6844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐿) = ( oppFunc ‘( oppFunc ‘𝐺)))
16	4	natrcl 17891	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐿𝑀𝐾) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
17	16	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
18	17	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 ∈ (𝑂 Func 𝑃))
19	14, 18	eqeltrrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐺) ∈ (𝑂 Func 𝑃))
20		relfunc 17800	. . . . . . 7 ⊢ Rel (𝑂 Func 𝑃)
21		eqid 2729	. . . . . . 7 ⊢ ( oppFunc ‘𝐺) = ( oppFunc ‘𝐺)
22	19, 20, 21	2oppf 49094	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐺)) = 𝐺)
23	15, 22	eqtr2d 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐺 = ( oppFunc ‘𝐿))
24	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 = ( oppFunc ‘𝐹))
25	24	fveq2d 6844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐾) = ( oppFunc ‘( oppFunc ‘𝐹)))
26	17	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 ∈ (𝑂 Func 𝑃))
27	24, 26	eqeltrrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))
28		eqid 2729	. . . . . . 7 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
29	27, 20, 28	2oppf 49094	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐹)) = 𝐹)
30	25, 29	eqtr2d 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 = ( oppFunc ‘𝐾))
31		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐿𝑀𝐾))
32	11, 12, 4, 13, 23, 30, 31	natoppf2 49192	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
33	1	2oppchomf 17661	. . . . . . . 8 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
34	33	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)))
35	1	2oppccomf 17662	. . . . . . . 8 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))
36	35	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐶) = (compf‘(oppCat‘𝑂)))
37	2	2oppchomf 17661	. . . . . . . 8 ⊢ (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃))
38	37	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃)))
39	2	2oppccomf 17662	. . . . . . . 8 ⊢ (compf‘𝐷) = (compf‘(oppCat‘𝑃))
40	39	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐷) = (compf‘(oppCat‘𝑃)))
41		natoppfb.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ 𝑉)
43		natoppfb.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ 𝑊)
45	1, 2, 42, 44, 27	funcoppc5 49107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 ∈ (𝐶 Func 𝐷))
46	45	func1st2nd 49038	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
47	46	funcrcl2 49041	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ Cat)
48	1	oppccat 17659	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
49	11	oppccat 17659	. . . . . . . 8 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat)
50	47, 48, 49	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑂) ∈ Cat)
51	46	funcrcl3 49042	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ Cat)
52	2	oppccat 17659	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
53	12	oppccat 17659	. . . . . . . 8 ⊢ (𝑃 ∈ Cat → (oppCat‘𝑃) ∈ Cat)
54	51, 52, 53	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑃) ∈ Cat)
55	34, 36, 38, 40, 47, 50, 51, 54	natpropd 17917	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐶 Nat 𝐷) = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
56	3, 55	eqtrid 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑁 = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
57	56	oveqd 7386	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐹𝑁𝐺) = (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
58	32, 57	eleqtrrd 2831	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹𝑁𝐺))
59	10, 58	impbida 800	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐹𝑁𝐺) ↔ 𝑥 ∈ (𝐿𝑀𝐾)))
60	59	eqrdv 2727	1 ⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Catccat 17601  Hom_f chomf 17603  comp_fccomf 17604  oppCatcoppc 17648   Func cfunc 17792   Nat cnat 17882   oppFunc coppf 49084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-cat 17605  df-cid 17606  df-homf 17607  df-comf 17608  df-oppc 17649  df-func 17796  df-nat 17884  df-oppf 49085

This theorem is referenced by:  fucoppclem  49369  lmddu  49629